intuitionistic logic

Brouwer's foundational theory of
mathematics which says that you should not count a proof of
(There exists x such that P(x)) valid unless the proof
actually gives a method of constructing such an x. Similarly,
a proof of (A or B) is valid only if it actually exhibits
either a proof of A or a proof of B.

In intuitionism, you cannot in general assert the statement (A
or not-A) (the principle of the excluded middle); (A or
not-A) is not proven unless you have a proof of A or a proof
of not-A. If A happens to be undecidable in your system
(some things certainly will be), then there will be no proof
of (A or not-A).

This is pretty annoying; some kinds of perfectly
healthy-looking examples of proof by contradiction just stop
working. Of course, excluded middle is a theorem of
classical logic (i.e. non-intuitionistic logic).